\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  -4.0 & -1.00 x_{1} &   & +  5.00 x_{3} & -7.00 x_{4} & -2.00 x_{5} & +  2.00 x_{6} & +  5.00 x_{7} & + 10.00 x_{8}\\
 x_{10}   &  -5.0 & +  8.00 x_{1} & +  4.00 x_{2} & -3.00 x_{3} & +  5.00 x_{4} & -7.00 x_{5} & +  8.00 x_{6} & +  9.00 x_{7} & +  4.00 x_{8}\\
 x_{11}   &  -21.0 & -5.00 x_{1} & +  5.00 x_{2} & -9.00 x_{3} & + 10.00 x_{4} & -2.00 x_{5} & +  6.00 x_{6} & +  3.00 x_{7} & +  7.00 x_{8}\\
 x_{12}   &  -1.0 & -1.00 x_{1} & +  2.00 x_{2} & +  1.00 x_{3} & -10.00 x_{4} & +  9.00 x_{5} & + 10.00 x_{6} & +  7.00 x_{7} & + 10.00 x_{8}\\
 x_{13}   &  14.0 & -10.00 x_{1} & -2.00 x_{2} & -4.00 x_{3} & +  4.00 x_{4} & +  3.00 x_{5} & + 10.00 x_{6} &   & +  1.00 x_{8}\\
 x_{14}   &  38.0 & -6.00 x_{1} &   & -6.00 x_{3} & -4.00 x_{4} & -1.00 x_{5} & +  3.00 x_{6} & +  8.00 x_{7} & -9.00 x_{8}\\
\hline
z    &  0.0 & -5.00 x_{1} & -5.00 x_{2} & -3.00 x_{3} & +  3.00 x_{4} & +  4.00 x_{5} & +  5.00 x_{6} &   & +  1.00 x_{8}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{8}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,14$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & +  1.00 y_{9} & -8.00 y_{10} & +  5.00 y_{11} & +  1.00 y_{12} & + 10.00 y_{13} & +  6.00 y_{14}\\
 y_{2}   &  1.0  &   & -4.00 y_{10} & -5.00 y_{11} & -2.00 y_{12} & +  2.00 y_{13} &   \\
 y_{3}   &  1.0 & -5.00 y_{9} & +  3.00 y_{10} & +  9.00 y_{11} & -1.00 y_{12} & +  4.00 y_{13} & +  6.00 y_{14}\\
 y_{4}   &  1.0 & +  7.00 y_{9} & -5.00 y_{10} & -10.00 y_{11} & + 10.00 y_{12} & -4.00 y_{13} & +  4.00 y_{14}\\
 y_{5}   &  1.0 & +  2.00 y_{9} & +  7.00 y_{10} & +  2.00 y_{11} & -9.00 y_{12} & -3.00 y_{13} & +  1.00 y_{14}\\
 y_{6}   &  1.0 & -2.00 y_{9} & -8.00 y_{10} & -6.00 y_{11} & -10.00 y_{12} & -10.00 y_{13} & -3.00 y_{14}\\
 y_{7}   &  1.0 & -5.00 y_{9} & -9.00 y_{10} & -3.00 y_{11} & -7.00 y_{12} &   & -8.00 y_{14}\\
 y_{8}   &  1.0 & -10.00 y_{9} & -4.00 y_{10} & -7.00 y_{11} & -10.00 y_{12} & -1.00 y_{13} & +  9.00 y_{14}\\
\hline
z    &  -0 & +  4.00 y_{9} & +  5.00 y_{10} & + 21.00 y_{11} & +  1.00 y_{12} & -14.00 y_{13} & -38.00 y_{14}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 7 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.60588235294 & -0.35 y_{8} & -0.25 y_{4} & -10.68 y_{10} & -0.76 y_{9} & +  8.64 y_{13} & + 10.19 y_{14}\\
 y_{2}   &  0.376470588235 & +  0.41 y_{8} & +  0.21 y_{4} & -1.29 y_{10} & +  2.64 y_{9} & +  3.26 y_{13} & -4.55 y_{14}\\
 y_{3}   &  2.04117647059 & -0.47 y_{8} & -0.57 y_{4} & -1.74 y_{10} & -5.71 y_{9} & +  1.25 y_{13} & + 12.52 y_{14}\\
 y_{7}   &  0.523529411765 & +  0.59 y_{8} & -0.11 y_{4} & -7.21 y_{10} & +  1.66 y_{9} & +  0.14 y_{13} & -12.85 y_{14}\\
 y_{5}   &  1.07647058824 & +  0.41 y_{8} & -0.49 y_{4} & +  6.21 y_{10} & +  9.54 y_{9} & -4.54 y_{13} & -0.75 y_{14}\\
 y_{11}   &  0.117647058824 & -0.06 y_{8} & -0.06 y_{4} & -0.53 y_{10} & -0.18 y_{9} & -0.29 y_{13} & +  0.76 y_{14}\\
 y_{6}   &  0.117647058824 & +  0.94 y_{8} & -0.06 y_{4} & -4.53 y_{10} & +  7.82 y_{9} & -9.29 y_{13} & -11.24 y_{14}\\
 y_{12}   &  0.0176470588235 & -0.06 y_{8} & +  0.04 y_{4} & -0.03 y_{10} & -0.88 y_{9} & +  0.11 y_{13} & +  0.36 y_{14}\\
\hline
z    &  2.48823529412 & -1.29 y_{8} & -1.19 y_{4} & -6.15 y_{10} & -0.58 y_{9} & -20.07 y_{13} & -21.58 y_{14}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  1.29411764706 & +  0.35 x_{1} & -0.41 x_{2} & +  0.47 x_{3} & -0.59 x_{7} & -0.41 x_{5} & +  0.06 x_{11} & -0.94 x_{6} & +  0.06 x_{12}\\
 x_{4}   &  1.19411764706 & +  0.25 x_{1} & -0.21 x_{2} & +  0.57 x_{3} & +  0.11 x_{7} & +  0.49 x_{5} & +  0.06 x_{11} & +  0.06 x_{6} & -0.04 x_{12}\\
 x_{10}   &  6.14705882353 & + 10.68 x_{1} & +  1.29 x_{2} & +  1.74 x_{3} & +  7.21 x_{7} & -6.21 x_{5} & +  0.53 x_{11} & +  4.53 x_{6} & +  0.03 x_{12}\\
 x_{9}   &  0.582352941176 & +  0.76 x_{1} & -2.64 x_{2} & +  5.71 x_{3} & -1.66 x_{7} & -9.54 x_{5} & +  0.18 x_{11} & -7.82 x_{6} & +  0.88 x_{12}\\
 x_{13}   &  20.0705882353 & -8.64 x_{1} & -3.26 x_{2} & -1.25 x_{3} & -0.14 x_{7} & +  4.54 x_{5} & +  0.29 x_{11} & +  9.29 x_{6} & -0.11 x_{12}\\
 x_{14}   &  21.5764705882 & -10.19 x_{1} & +  4.55 x_{2} & -12.52 x_{3} & + 12.85 x_{7} & +  0.75 x_{5} & -0.76 x_{11} & + 11.24 x_{6} & -0.36 x_{12}\\
\hline
z    &  -2.48823529412 & -1.61 x_{1} & -0.38 x_{2} & -2.04 x_{3} & -0.52 x_{7} & -1.08 x_{5} & -0.12 x_{11} & -0.12 x_{6} & -0.02 x_{12}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  1.29411764706 & +  0.35 x_{1} & -0.41 x_{2} & +  0.47 x_{3} & -0.59 x_{7} & -0.41 x_{5} & +  0.06 x_{11} & -0.94 x_{6} & +  0.06 x_{12}\\
 x_{4}   &  1.19411764706 & +  0.25 x_{1} & -0.21 x_{2} & +  0.57 x_{3} & +  0.11 x_{7} & +  0.49 x_{5} & +  0.06 x_{11} & +  0.06 x_{6} & -0.04 x_{12}\\
 x_{10}   &  6.14705882353 & + 10.68 x_{1} & +  1.29 x_{2} & +  1.74 x_{3} & +  7.21 x_{7} & -6.21 x_{5} & +  0.53 x_{11} & +  4.53 x_{6} & +  0.03 x_{12}\\
 x_{9}   &  0.582352941176 & +  0.76 x_{1} & -2.64 x_{2} & +  5.71 x_{3} & -1.66 x_{7} & -9.54 x_{5} & +  0.18 x_{11} & -7.82 x_{6} & +  0.88 x_{12}\\
 x_{13}   &  20.0705882353 & -8.64 x_{1} & -3.26 x_{2} & -1.25 x_{3} & -0.14 x_{7} & +  4.54 x_{5} & +  0.29 x_{11} & +  9.29 x_{6} & -0.11 x_{12}\\
 x_{14}   &  21.5764705882 & -10.19 x_{1} & +  4.55 x_{2} & -12.52 x_{3} & + 12.85 x_{7} & +  0.75 x_{5} & -0.76 x_{11} & + 11.24 x_{6} & -0.36 x_{12}\\
\hline
z    &  4.87647058824 & -3.89 x_{1} & -6.05 x_{2} & -0.82 x_{3} & -0.25 x_{7} & +  5.05 x_{5} & +  0.24 x_{11} & +  4.24 x_{6} & -0.06 x_{12}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  1.29411764706 & +  0.35 x_{1} & -0.41 x_{2} & +  0.47 x_{3} & -0.59 x_{7} & -0.41 x_{5} & +  0.06 x_{11} & -0.94 x_{6} & +  0.06 x_{12}\\
 x_{4}   &  1.19411764706 & +  0.25 x_{1} & -0.21 x_{2} & +  0.57 x_{3} & +  0.11 x_{7} & +  0.49 x_{5} & +  0.06 x_{11} & +  0.06 x_{6} & -0.04 x_{12}\\
 x_{10}   &  6.14705882353 & + 10.68 x_{1} & +  1.29 x_{2} & +  1.74 x_{3} & +  7.21 x_{7} & -6.21 x_{5} & +  0.53 x_{11} & +  4.53 x_{6} & +  0.03 x_{12}\\
 x_{9}   &  0.582352941176 & +  0.76 x_{1} & -2.64 x_{2} & +  5.71 x_{3} & -1.66 x_{7} & -9.54 x_{5} & +  0.18 x_{11} & -7.82 x_{6} & +  0.88 x_{12}\\
 x_{13}   &  20.0705882353 & -8.64 x_{1} & -3.26 x_{2} & -1.25 x_{3} & -0.14 x_{7} & +  4.54 x_{5} & +  0.29 x_{11} & +  9.29 x_{6} & -0.11 x_{12}\\
 x_{14}   &  21.5764705882 & -10.19 x_{1} & +  4.55 x_{2} & -12.52 x_{3} & + 12.85 x_{7} & +  0.75 x_{5} & -0.76 x_{11} & + 11.24 x_{6} & -0.36 x_{12}\\
\hline
z    &  4.87647058824 & -3.89 x_{1} & -6.05 x_{2} & -0.82 x_{3} & -0.25 x_{7} & +  5.05 x_{5} & +  0.24 x_{11} & +  4.24 x_{6} & -0.06 x_{12}\\
\end{array}\]


 $ x_{5} $ enters and $ x_{9} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  1.26896977175 & +  0.32 x_{1} & -0.30 x_{2} & +  0.22 x_{3} & -0.52 x_{7} & +  0.04 x_{9} & +  0.05 x_{11} & -0.60 x_{6} & +  0.02 x_{12}\\
 x_{4}   &  1.22393584207 & +  0.29 x_{1} & -0.35 x_{2} & +  0.86 x_{3} & +  0.03 x_{7} & -0.05 x_{9} & +  0.07 x_{11} & -0.34 x_{6} & +  0.00 x_{12}\\
 x_{10}   &  5.76804441703 & + 10.18 x_{1} & +  3.01 x_{2} & -1.98 x_{3} & +  8.29 x_{7} & +  0.65 x_{9} & +  0.41 x_{11} & +  9.62 x_{6} & -0.54 x_{12}\\
 x_{5}   &  0.0610734114744 & +  0.08 x_{1} & -0.28 x_{2} & +  0.60 x_{3} & -0.17 x_{7} & -0.10 x_{9} & +  0.02 x_{11} & -0.82 x_{6} & +  0.09 x_{12}\\
 x_{13}   &  20.3479333745 & -8.27 x_{1} & -4.51 x_{2} & +  1.47 x_{3} & -0.93 x_{7} & -0.48 x_{9} & +  0.38 x_{11} & +  5.57 x_{6} & +  0.31 x_{12}\\
 x_{14}   &  21.6224552745 & -10.13 x_{1} & +  4.34 x_{2} & -12.07 x_{3} & + 12.72 x_{7} & -0.08 x_{9} & -0.75 x_{11} & + 10.62 x_{6} & -0.30 x_{12}\\
\hline
z    &  5.18507094386 & -3.49 x_{1} & -7.44 x_{2} & +  2.21 x_{3} & -1.14 x_{7} & -0.53 x_{9} & +  0.33 x_{11} & +  0.09 x_{6} & +  0.40 x_{12}\\
\end{array}\]


 $ x_{3} $ enters and $ x_{14} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  1.67024539877 & +  0.13 x_{1} & -0.22 x_{2} & -0.02 x_{14} & -0.28 x_{7} & +  0.04 x_{9} & +  0.04 x_{11} & -0.41 x_{6} & +  0.02 x_{12}\\
 x_{4}   &  2.77044989775 & -0.43 x_{1} & -0.04 x_{2} & -0.07 x_{14} & +  0.94 x_{7} & -0.06 x_{9} & +  0.01 x_{11} & +  0.42 x_{6} & -0.02 x_{12}\\
 x_{10}   &  2.21625766871 & + 11.85 x_{1} & +  2.30 x_{2} & +  0.16 x_{14} & +  6.20 x_{7} & +  0.66 x_{9} & +  0.54 x_{11} & +  7.88 x_{6} & -0.49 x_{12}\\
 x_{5}   &  1.13445807771 & -0.42 x_{1} & -0.06 x_{2} & -0.05 x_{14} & +  0.46 x_{7} & -0.11 x_{9} & -0.02 x_{11} & -0.29 x_{6} & +  0.08 x_{12}\\
 x_{13}   &  22.9877300613 & -9.51 x_{1} & -3.98 x_{2} & -0.12 x_{14} & +  0.62 x_{7} & -0.49 x_{9} & +  0.29 x_{11} & +  6.86 x_{6} & +  0.28 x_{12}\\
 x_{3}   &  1.79192229039 & -0.84 x_{1} & +  0.36 x_{2} & -0.08 x_{14} & +  1.05 x_{7} & -0.01 x_{9} & -0.06 x_{11} & +  0.88 x_{6} & -0.02 x_{12}\\
\hline
z    &  9.1436605317 & -5.34 x_{1} & -6.65 x_{2} & -0.18 x_{14} & +  1.19 x_{7} & -0.54 x_{9} & +  0.19 x_{11} & +  2.03 x_{6} & +  0.35 x_{12}\\
\end{array}\]


 $ x_{6} $ enters and $ x_{5} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  0.0993204391009 & +  0.72 x_{1} & -0.13 x_{2} & +  0.05 x_{14} & -0.91 x_{7} & +  0.19 x_{9} & +  0.06 x_{11} & +  1.38 x_{5} & -0.09 x_{12}\\
 x_{4}   &  4.38525875588 & -1.04 x_{1} & -0.12 x_{2} & -0.14 x_{14} & +  1.59 x_{7} & -0.21 x_{9} & -0.01 x_{11} & -1.42 x_{5} & +  0.09 x_{12}\\
 x_{10}   &  32.6734622757 & +  0.48 x_{1} & +  0.67 x_{2} & -1.17 x_{14} & + 18.46 x_{7} & -2.26 x_{9} & +  0.03 x_{11} & -26.85 x_{5} & +  1.58 x_{12}\\
 x_{6}   &  3.86652726956 & -1.44 x_{1} & -0.21 x_{2} & -0.17 x_{14} & +  1.56 x_{7} & -0.37 x_{9} & -0.06 x_{11} & -3.41 x_{5} & +  0.26 x_{12}\\
 x_{13}   &  49.5291862694 & -19.41 x_{1} & -5.40 x_{2} & -1.28 x_{14} & + 11.30 x_{7} & -3.03 x_{9} & -0.15 x_{11} & -23.40 x_{5} & +  2.08 x_{12}\\
 x_{3}   &  5.19411047221 & -2.11 x_{1} & +  0.18 x_{2} & -0.23 x_{14} & +  2.42 x_{7} & -0.33 x_{9} & -0.12 x_{11} & -3.00 x_{5} & +  0.21 x_{12}\\
\hline
z    &  17.0054016379 & -8.27 x_{1} & -7.07 x_{2} & -0.53 x_{14} & +  4.36 x_{7} & -1.30 x_{9} & +  0.06 x_{11} & -6.93 x_{5} & +  0.88 x_{12}\\
\end{array}\]


 $ x_{7} $ enters and $ x_{8} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  0.108820160367 & +  0.79 x_{1} & -0.15 x_{2} & +  0.05 x_{14} & -1.10 x_{8} & +  0.21 x_{9} & +  0.07 x_{11} & +  1.52 x_{5} & -0.10 x_{12}\\
 x_{4}   &  4.5578465063 & +  0.21 x_{1} & -0.35 x_{2} & -0.05 x_{14} & -1.74 x_{8} & +  0.12 x_{9} & +  0.10 x_{11} & +  0.98 x_{5} & -0.07 x_{12}\\
 x_{10}   &  34.6823214968 & + 15.01 x_{1} & -2.03 x_{2} & -0.15 x_{14} & -20.23 x_{8} & +  1.63 x_{9} & +  1.31 x_{11} & +  1.16 x_{5} & -0.27 x_{12}\\
 x_{6}   &  4.03589156166 & -0.22 x_{1} & -0.43 x_{2} & -0.08 x_{14} & -1.71 x_{8} & -0.04 x_{9} & +  0.04 x_{11} & -1.05 x_{5} & +  0.11 x_{12}\\
 x_{13}   &  50.7590683467 & -10.52 x_{1} & -7.05 x_{2} & -0.66 x_{14} & -12.38 x_{8} & -0.65 x_{9} & +  0.63 x_{11} & -6.25 x_{5} & +  0.95 x_{12}\\
 x_{3}   &  5.45780832379 & -0.20 x_{1} & -0.18 x_{2} & -0.10 x_{14} & -2.66 x_{8} & +  0.18 x_{9} & +  0.05 x_{11} & +  0.68 x_{5} & -0.04 x_{12}\\
\hline
z    &  17.4795723559 & -4.84 x_{1} & -7.71 x_{2} & -0.29 x_{14} & -4.77 x_{8} & -0.38 x_{9} & +  0.36 x_{11} & -0.32 x_{5} & +  0.44 x_{12}\\
\end{array}\]


 $ x_{11} $ enters and Unbounded Dictionary!


 $ x_{11} $ enters and Unbounded Dictionary!
\end{document}
